Quantum stereographic projection and the homographic oscillator

Quantum stereographic projection and the homographic oscillator

T. Hakiog˘lu

Physics Department, Bilkent University, Ankara 06533 Turkey M. Arik

Physics Department, Bog˘azic¸i University, Istanbul 80815 Turkey ~Received 12 October 1995!

The quantum deformation created by the stereographic mapping from S2toC is studied. It is shown that the resulting algebra is locally isomorphic to su~2! and is an unconventional deformation of which the undeformed limit is a contraction onto the harmonic oscillator algebra. The deformation parameter is given naturally by the central invariant of the embedding su~2!. The deformed algebra is identified as a member of a larger class of quartic q oscillators. We next study the deformations in the corresponding Jordan-Schwinger representation of two independent deformed oscillators and solve for the deforming transformation. The invertibility of this transformation guarantees an implicit coproduct law which is also discussed. Finally we discuss the analogy between Poincare´’s geometric interpretation of the quantum Stokes parameters of polarization and the stereo-graphic projection as an important physical application of the latter.@S1050-2947~96!00106-0#

PACS number~s!: 03.65.Fd, 02.20.Sv, 02.20.Qs

I. INTRODUCTION

Since the discovery of the first deformed quantum algebra by Biedenharn and Macfarlane @1# a tremendous effort has been made to find physical realizations of these algebras. Much earlier, inspired by an operator representation for

q-deformed dual resonance models@2#, Baker, Coon, and Yu

formulated the simplest q algebra which produced a suitably bounded spectrum determined by the parameter of deforma-tion q. Apart from their profound mathematical significance mainly related to the solution of the quantum Yang-Baxter equation @3#, and other problems @4#, physical applications can be found in the deformed Jaynes-Cummings model@5#, the ubiquitous quantum phase problem @6#, the relativistic

q oscillator @7#, and recently, in reproducing deformed

nuclear energy levels @8#, the Morse oscillator @9#, and the Kepler problem@10#.

In particular, several possible realizations of deformed Lie algebras can be constructed by applying certain nonlinear invertible deforming transformations to the generators of the undeformed algebras @11#. In this context, some explicit cases have been examined by Curtright and Zachos @12#, among several previously studied examples.

The quantum deformation of a physical symmetry should be identified by a deformation parameter which must be uniquely determined by a set of observables. In this work, we give a particular example of that by examining the ste-reographic projection of the su~2! generators on an extended complex plane and show that the resulting deformation is described by a deformation parameter which is directly con-nected with the central invariant of the embedding su~2!. In Sec. II we define and derive the properties of the quantum stereographic projection. Section III is devoted to the prop-erties of the central invariant. In Sec. IV the deformation induced by the homographic oscillator on su~2! is studied using Jordan-Schwinger representation. The proof of exist-ence of the coproduct for the corresponding su~2! deforma-tion is presented in Sec. V. In the last secdeforma-tion, Sec. VI, we

discuss possible physical applications where quantized ste-reographic projection and the resulting homographic oscilla-tor algebra become relevant.

II. STEREOGRAPHIC PROJECTION

Stereographic projection~SP! is a mapping from the Rie-mann sphere S2 onto an extended complex plane C. At the

classical level, SP is defined by a mapping from S2in

spheri-cal J,u,fparametrization to that z,z*on the complex plane given by z52Jcotu/2 eif, with S_u5 2v 11v2, C_u5v 2_21, v2₁₁, where v5

A

z*z 2J , ~1!

where J is fixedu andf are real coordinates describing the polar and azimuthal angles, S_u and C_u describe the sine and cosine of u, respectively, and z,z* are defined on the pro-jected plane as shown in Fig. 1. SP is an invertible mapping except for the ideal point at infinity. However, this does not violate the formal equivalence between the two representa-tions; since, as will be shown later, when Eq. ~1! is quan-tized, the ideal point at infinity is well represented by infinity in the discrete spectrum of the corresponding deformed alge-bra. We now proceed by defining an operator realization of Eq. ~1! via sine-cosine operators Cˆ_u and Sˆ_u as

Cˆ_u5Vˆ 2₂₁ Vˆ2₁₁, Sˆu5 2Vˆ 11Vˆ2, where Vˆ5

A

Zˆ†Zˆ, Zˆ5Eˆ_fVˆ, ~2! 54

and Vˆ and Eˆ_f are operator counterparts of v and eif, re-spectively. As in the case of angular decomposition of su~2!, here we deal with ideal unitary polar operators ~i.e.,

Cˆ_u21Sˆ_u251 and @Cˆ_u,Sˆ_u#50) whereas the azimuthal phase operator Eˆ_f has both a nonunitary as well as a unitary rep-resentation.

Throughout this work we assume that a quantum defor-mation is understood explicitly in the same sense as Refs.

@11,12#. To be more precise, providing invertible nonlinear

functionals of generators, the deformed algebraic structure and its representations are obtained by directly applying these nonlinear functionals to a particular representation of the undeformed mother algebra su~2! @11#. Depending on which level this substitution and following quantization take place, one naturally obtains different quantum deformations.

Using Eqs.~2! it can be found that

Zˆ†Zˆ511Cˆu 12Cˆ_u, ZˆZˆ †_5Eˆ f 11Cˆ_u 12Cˆ_uE ˆ f †_{. ~3!}

In deriving Eq.~3! we did not assume the existence of any particular unitary representation for Eˆ_f~i.e., Eˆ_f†ÞEˆ_f21). The algebraic structure of the Zˆ,Zˆ†operators, however, as will be discussed later, is not influenced by any conflict between the unitary and nonunitary description of Eˆ_f. The algebra de-fined by Zˆ,Zˆ† can be found by using the well-known su~2! relation@13,14# @Cˆu,Eˆf#52 1 JE ˆ f ~4!

in Eq. ~3!, which is expressed by the generalized

commuta-tion relacommuta-tion

aˆaˆ†5paˆ †_aˆ11

kaˆ†aˆ11~11@Eˆf,E ˆ f †_{#!, with aˆ5}1 aZˆ, aˆ†5 1 a*Zˆ†, ~5!

where aˆ(aˆ†) represent the normalized annihilation~creation! operators and the parameters q,k,a are given by

uau2_{5u2J21u21, p5~2J11!/~2J21!,}

k52~2J21!22. ~6!

Equation~5! is not an example of prototype deformed alge-bras and has not been studied in the literature. The commu-tation@Eˆ_f,Eˆ_f†# naturally arises in the derivation. It has been

recently shown that it is possible to find a manifestly unitary description of Eˆ_f without affecting any of its operator prop-erties@6#. Here, Eˆ_fis identical to the azimuthal phase opera-tor in the su~2! polar decomposition @6,15# and its unitary representation has the cyclic property

Eˆ_f5

(

m_{52 j} j

u jm21

&^

j mu1bu j2 j

&^

j ju, @Eˆ_f,Eˆ_f†#50

~7!

in the finite dimensional Hilbert space spanned by the basis vectors u jm

&

. Here ubu51 and is otherwise undetermined, referring to an arbitrary reference phase. On the other hand, we must mention as a side remark that, if one adopts the nonunitary description of Eˆ_f @i.e. b50 in ~7!#, the first de-formed excitation energy of the algebra ~5! is influenced by the nonunitarity of Eˆ_f in such a way that it produces a scale transformation on the operators aˆ,aˆ†. However, its effect can always be absorbed by a further trivial renormalization of these operators. We will not elaborate on the other implica-tions of the nonunitary description of the Eˆ_f operator in this work.

Equation~5! is actually in the class of generalized quartic oscillators@16# whose properties cannot be simply obtained by taking the square of any generalized commutator. Be-cause of the homographic dependence of aˆ†aˆ on aˆaˆ† we term the algebra in Eq.~5! a homographic oscillator ~HO!. In the limit J→` we observe the limits p→1 and k→0, there-fore HO contracts to the simple harmonic oscillator ~SHO!. This is reminiscent of the I˙no¨nu¨-Wigner contraction of su~2! onto the SHO @17#. The spectrum of HO can be solved ex-actly by considering a generalized Hermitian number opera-tor Nˆ such that @aˆ,Nˆ#5aˆ;@aˆ†,Nˆ #52aˆ†. For the most gen-eral solution, we have aˆ†aˆ5@Nˆ# and aˆaˆ†5@Nˆ11# where @Nˆ# is the principal number operator, and un

&

J are the basis

vectors such that

@n11#5p_k@n#11_@n#11, ~8!

where

aˆun

&

J5@n#1/2un21

&

J, aˆ†un

&

J5@n11#1/2un11

&

J,

Nˆ un

&

J5nun

&

J. ~9!

Enforcing the ground state u0

&

J to be annihilated by aˆ, we

have@0#50. Using this ground state in ~8!, the whole spec-trum can be analytically iterated to yield

@n#5_{@@n##2@@n21##}@@n## , @@n##5r1 n_2r 2 n r12r2 ~r 1Þr2!, ~10!

where@0#5@@0##50 and @1#5@@1##51, with 1 r11 1 r2511p, 1 r1r25p2k. ~11!

From Eqs.~6! and ~11! we find r15r25q. The spectrum is thus given by the first derivative of@@n## with respect to q as FIG. 1. Geometric interpretation of stereographic projection.

@@n##5nqn21_{. Here we identify q5(2J21)/2J as the}

de-formation parameter of the HO algebra.

It is known that the basic number @@n## arises in the so-lution of the generalized Fibonacci series@16#

@@n12##5a@@n11##1b@@n##,

where

a5r11r2, b52r1r2. ~12!

Further, it can be shown that Eq. ~12! defines a class of

generalized~i.e., r1Þr221) Biedenharn-Macfarlane~BM! ~or

Fibonacci! q oscillator. The commutation relation which yields ~12! can be found if two new operators bˆ,bˆ† are de-fined such that bˆ†bˆ5@@Nˆ##,bˆbˆ†5@@Nˆ11## as

bˆbˆ†5r2bˆ†bˆ1r1Nˆ. ~13!

In principle, Eqs.~10!, ~12!, and ~13! plausibly suggest that Eq.~5! can be effectively obtained from the generalized BM

q oscillator by a second deformation. Although a direct

transformation from one into the other has not been found, recently an attempt has been made to unify all quartic oscil-lators. In this scheme, ~5! and ~13! correspond to special cases such as r25r121, r25r₁2, or r25r₁ ~see Ref. @18#!. This can be shown by applying the nonlinear transformation

bˆ5b˜ˆ ~b1g˜bˆ˜bˆ†!1/2, ~14! where@b˜ˆ,Nˆ #5b˜ˆ in Eq. ~13!, we get the form

Ab˜ˆ˜bˆ†˜ˆb˜bˆ†1Bb˜ˆ˜bˆ†˜bˆ†˜bˆ 1Cb˜ˆ†˜bˆb˜ˆ†b˜ˆ

1Db˜ˆ˜bˆ†_1Eb˜ˆ†˜_bˆ 1F50, ~15!

with the coefficients

A5g, B50 , C52qg,

D5b, E52qb, F521, ~16!

whereas the HO corresponds to the special case of the gen-eralized quartic oscillator in Eq.~15! with the coefficients

A50 , B5k, C50, D51 , E52q, F521.

~17!

Both Eqs.~16! and ~17! have the property AE5CD which is possessed by the quartic square root oscillator@16# as a spe-cial case of ~15!. In the limit J→` both homographic and Fibonacci oscillators contract to SHO.

III. CENTRAL INVARIANT

The HO algebra in ~5! is isomorphic to its underlying su~2! algebra. The range of values which the quantum num-ber n can take is limited by the total angular momentum J

~i.e., 0<n<2 J). This can be seen easily by causing the

diagonal operator Cˆ_u to act on the angular momentum

uJm

&

and, simultaneously, on the homographic oscillator

un

&

J bases. The action of the aˆ,aˆ

† _{operators on the basis}

vectors generates lower and upper bounds for its energy spectrum ~i.e., aˆu0

&

_J5aˆ†u2J

&

_J50). By direct substitution we find@2 J#5`.

In order to find the central invariant Cˆq we first write the

HO algebra in the conventional form

@aˆ,Nˆ#5aˆ, @aˆ†_{,Nˆ #52aˆ}†_{, @aˆ,aˆ}†_#5G

q~Nˆ!, where Gq~Nˆ!5 Q~11Q! ~Nˆ2Q!~Nˆ212Q!, Q5q/~12q!. ~18! Cˆ_q is then formulated as

Cˆ_q5aˆ†_aˆ1aˆaˆ†_1F~Nˆ!,

where

F~Nˆ!5 SN

ˆ2_1TNˆ1U

~Nˆ2Q!~Nˆ212Q! ~19!

with the coefficients S5(cq12 Q), T5@2cq(112 Q)

22 Q2_{], and U5(c}

q21)(Q1Q2). Here cq is an

undeter-mined eigenvalue of Cˆq. It is easy to see also from Eqs.~18!

and ~19! that there are lower (n50) and upper (n52 J) bounds in the spectrum such that

aˆ†aˆu0

&

J5aˆaˆ†u2J

&

J50

and

F~2J!2Gq~2J!5F~0!1Gq~0!. ~20!

IV. THE HOMOGRAPHIC q-BOSON REALIZATION OF su„2… DEFORMATION

In the q-boson realization of su~2! deformation, the fun-damental spinor realization is mapped onto a pair of com-muting homographic oscillators as

Iˆ₁5aˆ1†aˆ2, Iˆ25aˆ1aˆ2 †

, Iˆz5

1

2~Nˆ12Nˆ2!, ~21! where independent algebras for aˆ₁ and aˆ₂ are given by the analogs of~5!. Using ~5! and ~6! the operators in ~21! can be found to satisfy

@Iˆ6,Iˆz#57Iˆ6, @Iˆ1,Iˆ2#5 fIˆ~Iˆz!2 fIˆ~Iˆz21!, ~22!

where fIˆ~Iˆz!5Q1Q2 Iˆ2Iˆ_z Iˆ2Iˆ_z212Q2 Iˆ1Iˆ_z11 Iˆ1Iˆ_z2Q₁. ~23!

Here Q_i5q_i/(12q_i) (i51,2) and q_i’s are the deformation parameters. It is also easy to see that Iˆ2_51/2(Iˆ

1Iˆ2

1Iˆ₂Iˆ₁)1Iˆ_z2 is an invariant of the algebra with eigenvalue

i(i11) where i51/2(n11n₂) and i_z51/2(n12n2). Hence

As q1 and q2 independently approach unity in the zero

de-formation limit, the deformed algebra in ~22! approaches su~2!.

Now, our aim is to find the invertible nonlinear transfor-mation between the generators Iˆ₆,Iˆz and the generators

Jˆ₆,Jˆz of the limiting su~2!. We seek an invertible operator

function G (Jˆz) such that@11#

Iˆ₁5G ~Jˆz!Jˆ1, Iˆ25Iˆ₁

†

, Iˆz5Jˆz1const, ~24!

where the constant only depends on the central invariant. The type of deformation in ~22! is not suitable for any Laplace~or Fourier! representation @11# in terms of the pow-ers of Jˆz. However, one can still find the central element Cˆ of this algebra such that

Iˆ₁Iˆ₂5G2~Jˆz!Jˆ1Jˆ25Cˆ 2F ~Jˆz!,

Iˆ₂Iˆ₁5Jˆ₂G2~Jˆz!Jˆ15Cˆ 2F ~Jˆz11!,

~25!

where F (Jˆz) is to be found. Since Iˆ is the group invariant, Cˆ is a function of Iˆ only. Therefore its eigenvalue cQ₁Q₂

only depends on Iˆ and the deformation parameters q1,q2.

Furthermore, the existence of the lowermost and uppermost states on which the action of Iˆ₂ and Iˆ₁yields zero, respec-tively, implies that Cˆ5F (2J)5F (J11). Using the con-dition@Iˆ₆,Cˆ#50, the operator function F (Jˆz) can be easily

found as

F~Jˆz!52 fIˆ~Jˆz21!. ~26!

From ~24!, ~25!, and the su~2! relation Jˆ₁Jˆ₂ 5 1 2(Jˆ 2_2Jˆ z 2_1Jˆ z) we finally obtain G2~Jˆz!52 fIˆ~Jˆz21!1Cˆ ~Jˆ2_2Jˆ z 2_1Jˆ z! . ~27!

Equations~23!, ~26!, and ~27! define the invertible deforma-tion G (Jˆz).

V. COPRODUCT

One implication of Sec. IV is that the existence of the simple su~2! coproduct

D~Jˆ6!5Jˆ6^I1I^Jˆ6,

D~Jˆz!5Jˆz^I1I^Jˆz, ~28!

and the invertibility of G (Jˆz) guarantee a coproduct law@11#

for Iˆ₆,Iˆz. The deforming map defined in Eqs.~24! and ~27!

is, however, not in the class of generalized prototype su(2)q deformations of Curtright and Zachos @11,12#. The

nonpolynomial forms of Gq(Nˆ ) and fIˆ(Iˆz) do not permit a

closed analytic form for the coproduct D(Iˆ₆) and D(Iˆz).

However, one can get some hint from the interesting sym-metry displayed by Eq.~22! in the limits of very large ~i.e.,

qi→2`) and very small ~i.e., qi→1) deformations as

limq_i→1@Iˆ1,Iˆ2#52 Iˆz,

limq_i→2`@Iˆ₁,Iˆ₂#54eIˆz/Iˆ,

where

Qi52~11e!, e!1. ~29!

Hence in both limits the deformed algebra ~22! behaves like a pure su~2!. In principle, D(Iˆ₆) andD(Iˆz) can be obtained

by the application of the deforming invertible transformation found in Eqs.~24! and ~27! ~e.g., see Ref. @11#! as

D~Gˆ!5T@I^T21~Gˆ!1T21~Gˆ!^I#, ~30!

where gˆ5(Jˆ₆,Jˆz), Gˆ 5(Iˆ6,Iˆz), and Gˆ 5T(gˆ) is just a

com-pact notation for the transformation in Eq. ~24!. Equation

~30! implies

@D~Iˆ6!,D~Iˆz!#57D~Iˆ6!, @D~Iˆ1!,D~Iˆ2!#5D~Iˆz!.

~31!

Here we notice that D(Gˆ) should behave like D(gˆ) in the symmetric limits ~i.e., qi→1,qi→2`). A possibly existing

simple analytic form of D(Gˆ) might be connected to the closed form~30! by a unitary transformation @12#. However, no explicit and general form for such a transformation is known at the moment.

VI. DISCUSSION

The isomorphism between the homographic oscillator al-gebra ~5! and su~2! has subtle implications in the angular momentum addition theorem and the coproduct law for

Iˆ₆,Iˆz. From ~24! and ~25! it is easy to see that

@Iˆ,Jˆ6#5@Iˆ,Jˆz#50 and @Iˆ,Jˆ2#50. These commutations

fur-ther imply an isomorphism betweenuii_z

&

andu j j_z

&

. In other words, the two basis vectors are parallel to each other al-though they are raised and lowered in different scales on the

z axis@19#. We also note that Iˆ is a function of Jˆ only. Let us

now define i and i_z as quantum numbers corresponding to a basis set uiiz

&

on which the generators in~22! apply. Then,

using ~27! and acting Eq. ~24! on this basis, one obtains

j~ j11!5cQ₁Q₂1Q1Q2

i~i11!

~i2Q1!~i2Q221!, ~32!

which is the desired relation between Iˆ and Jˆ. Here c_Q

1Q2 is

the eigenvalue of Cˆ . In the undeformed limit ~i.e.,

Q₁,Q2→`) the equivalence of the two algebras requires

cQ₁Q₂→0.

The arguments presented above guarantee the existence of an implicit coproduct, making it possible to consider Iˆ₆and

Iˆz as elements of a quantum deformation of su~2!. The

de-formation parameter is shown to be determined by the total

angular momentum J. Our work is under progress to extend

the arguments presented above to the most general case of quartic oscillator algebra. To the knowledge of the authors, such generalized cases have also been examined recently by Smith @20# as applied to the more general nonunitary case

The stereographic projection is intimately connected with the polar construction of su~2! generators, the quantum phase problem, and, in particular, certain geometrical realizations

@21# of quantum Stokes parameters of polarization @14,21,22#. Nevertheless, the nonunitarity of the azimuthal

phase Eˆ_fand/or the unavoidable nonzero commutations be-tween the azimuthal and polar phase operators@e.g., see Eq.

~4! here# plague the polar operator construction of su~2!. As

briefly mentioned in Sec. II, the resolution was given by Ellinas@6# by adding a cyclic property to the matrix elements of Eˆ_f along the Jz axis with a periodicity of (2J11). This

new term does not affect the original spectrum and further makes Eˆ_f manifestly unitary.

The homographic oscillator representation is physically relevant for its application, particularly in the weak intensity regime of quantum Stokes parameters@14,21#. The quantum phase problem has been studied in the context of an opera-tional point of view by Noh, Fouge´res, and Mandel using two coherent laser sources @23#. More recently this formal-ism has been applied to the case of polarization measurement

of weak fields @14#. Using Poincare´’s stereographic projec-tion, the angular parameters of the polarization ellipse can be mapped conveniently on the Stokes parameters. This has a certain advantage from the operational point of view. The direct measurement of the quantum Stokes parameters might be more relevant in determining the orientation of the polar-ization ellipse both for experimental perspective and the suit-able group properties that quantum Stokes parameters pos-sess. This is where the authors believe that the homographic oscillator introduced here can be linked with the quantum Stokes parameters and polarization measurement. Another dimension of our work in progress is to exploit this physical application.

ACKNOWLEDGMENTS

T. H. appreciates critical comments by Professor C. K. Zachos and Professor S. Stepanov. He is also grateful to Professor O. Viskov for remarks on Sec. III and for bringing Ref. @20# to his attention.

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